Distributed quantum sensing of multiple phases with fewer photons

Distributed quantum metrology has drawn intense interest as it outperforms the optimal classical counterparts in estimating multiple distributed parameters. However, most schemes so far have required entangled resources consisting of photon numbers equal to or more than the parameter numbers, which is a fairly demanding requirement as the number of nodes increases. Here, we present a distributed quantum sensing scenario in which quantum-enhanced sensitivity can be achieved with fewer photons than the number of parameters. As an experimental demonstration, using a two-photon entangled state, we estimate four phases distributed 3 km away from the central node, resulting in a 2.2 dB sensitivity enhancement from the standard quantum limit. Our results show that the Heisenberg scaling can be achieved even when using fewer photons than the number of parameters. We believe our scheme will open a pathway to perform large-scale distributed quantum sensing with currently available entangled sources.

multiple unknown phases has been limited by the difficulty associated with generating multi-photon entangled states 24 .
In this work, we propose a distributed quantum sensing protocol that achieves quantum-enhanced sensitivity even when the number of photons used is less than the number of unknown phases to estimate.We also experimentally demonstrate distributed quantum-enhanced detection of the average of four phases located in distant nodes over 3 km away from the central node with two-photon entangled states.To this end, we distribute the prepared two-photon entangled state among four nodes.Then the probe states undergo phase encoding at each node, which is separated by 3 km away from the central node through optical fiber spools, and are measured by four local measurements.We estimate an average of four unknown phases with the maximum likelihood estimator (MLE), and achieved a sensitivity beyond the SQL.We believe that our results provide a useful platform to investigate a distributed quantum sensor network.

Results
Let us begin by introducing a general scenario of distributed quantum sensing to estimate the spatially distributed d multiple phases as shown in Fig. 1 8,16 .There are d unknown phases ϕ = (ϕ 1 , ϕ 2 ,…, ϕ d ), and they are distributed in different locations.In a distributed quantum sensing scenario, the goal is to estimate a linear global function of φ = α T ϕ, where α = (α 1 , α 2 , …, α d ) and P d j = 1 jα j j = 1.Then the unitary phase evolution is given by ÛðϕÞ = expðÀi P d j = 1 Ĥj ϕ j Þ = expðÀi Ĥ Á ϕÞ, where Ĥ = ð Ĥ1 , . . ., Ĥd Þ denotes the local Hamiltonians.The initial probe state Ψ j i interacts with phase encoding ÛðϕÞ and evolves into ÛðϕÞ Ψ j i.Finally, it is detected by a set of projectors f Πl g, and it gives a corresponding probability set {P l } for a given ϕ.Then, we can estimate φest from {P l } using a proper estimator, for instance, MLE 21,22,25 .Here, the uncertainty bound for the estimation of φ can be described as follows: which is known as the weak form of Cramer-Rao bound (CRB), F denotes the Fisher information matrix with elements F (j, k) = ∑ l (1/P l ) (∂P l /∂ϕ j )(∂P l /∂ϕ k ), and μ denotes the number of measurements 16 .
Hereafter, we will simplify the description with μ = 1 for all the presented theoretical schemes.
To achieve the HS (Δ 2 φ = 1=N 2 ), the MePe state, which was proposed in refs.16 and 21 , has the form of where H(V) represents a horizontal (vertical) polarization, N denotes the total number of photons, d denotes the number of unknown phases (i.e., number of nodes), and N/d is the number of photons in each node j = 1, 2, ..., d.In Eq. ( 2), the photon number N should be equal to or larger than the number of unknown phases d.The MePe state requires the photon number N = m × d (m is a positive integer).Thus, to achieve the HS using a MePe state in distributed sensing, at least dphoton GHZ-like entangled states are required.However, a key challenge in the approach is the difficulty in generating d-photon MePe states for large d 24 .
Here, instead, we consider another probe state that can achieve the HS without the constraint N ≥ d.The probe state we propose has the form of where d + 1 ≡ 1 since node d is adjacent to node 1 as shown in Fig. 1, for instance, . Equation (3) corresponds to a coherent superposition of the N-photon two-mode polarizationentangled states in which N/2 photons are distributed in two adjacent nodes, respectively.In this strategy, the photon number in each node is N/2, so the only condition of our probe state is that N is an even number.It means that our probe state can be used to estimate d unknown phases with N photons even though N < d.By adding a beam splitter network (BSN) to multiplex the state, we are able to simultaneously probe multiple paths.In our experiment, we consider a two-photon probe state Ψ 2 4

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to estimate four unknown phases, i.e., N = 2 and d = 4.
For demonstration purposes, we experimentally estimate an average of four spatially distributed phases φ = P 4 j = 1 ϕ j =4 (see Fig. 2a), and ϕ j corresponds to the phase encoding at node j.Note that four sensors are distributed 3 km away from the central node, respectively.To this end, we prepared the two-photon polarization entangled probe states in Eq. ( 3) with N = 2 and d = 4 distributed in four nodes as shown in Fig. 2b.Firstly, the polarization Bell state was generated via spontaneous parametric down-conversion (SPDC) process from a 10-mm-thick type-II periodically-poled KTiOPO 4 (PPKTP) crystal, which is located at the Sagnacinterferometer 26,27 .Then, the generated Bell state was split to distribute the quantum states among four nodes by the BSN consisting of two 50/50 beam splitters as shown in Fig. 2b.The process of preparing Ψ 2 4 E can be described as follows: where the subscript denotes the node, for example, jΦ j,k i denotes the Bell states ðjH j H k i + jV j V k iÞ= ffiffiffi 2 p between the node j and the node k.Then, Ψ 2 4

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can be interpreted as a superposition of four Bell states between the two adjacent nodes such as jΦ 1,2 i, jΦ 2,3 i, Φ 3,4 , and Φ 4,1 . Then, the probe state jΨ 2 4 i evolved into ÛðϕÞjΨ 2 4 i after the phase encoding.The phase encoding was realized by a set of quarter waveplate (QWP), half waveplate (HWP), and QWP located at each node, as shown in Fig. 2a 26,27 .
Then ÛðϕÞ Ψ 2 4 E was measured by projective measurement on the σ x basis at each node 21,22 .The σ x measurement was performed with a combination of an HWP at 22.5 ∘ from the optic axis and a polarizing beam splitter (PBS), and the coincidence measurement in the σ x basis was performed in each node.Then, it gives a corresponding probability set {P l }, and there are sixteen probabilities of fP + + jk ,P + À jk ,P À + jk ,P ÀÀ jk g where the superscripts of + and − correspond to the outcomes of the σ x measurement, and jk denotes the nodes and depending on ϕ j and ϕ k , i.e., {jk} = {12, 23, 34, 41}.For example, P + À 12 denotes the outcome probability of two-photon coincidence events between + in node 1 and − in node 2. The outcome probabilities in our experiment can be written as where V ± ± jk ( V ± ∓ jk ) is the visibility of P ± ± jk (P ± ∓ jk ).The sensitivity bound of our probe state can be calculated using Eq. ( 5) and reach the HS beyond the SQL.See Methods for the detailed calculation.
Our experimental results on fP + + 12 ,P + À 12 ,P À + 12 ,P ÀÀ 12 g by scanning ϕ 1 and ϕ 2 are shown in Fig. 3, and their averaged visibilities are obtained to be 0.955, 0.981, 0.970, and 0.945, respectively.Note that P 12 , P 23 , P 34 and P 41 are the outcome probabilities with respect to {ϕ 1 , ϕ 2 }, {ϕ 2 , ϕ 3 }, {ϕ 3 , ϕ 4 }, and {ϕ 4 , ϕ 1 }, respectively.We provide our experimental results on other probabilities in Supplementary Notes.Then, we estimate an average of unknown four phases, i.e., φ, using MLE 21,22 .To obtain the sensitivity for estimating φ, we calculated the standard deviation of the estimated average of four phases, i.e., Δ φest .To this end, the sixteen probabilities {P l } were obtained by scanning ϕ 1 when ϕ 2 , ϕ 3 , and ϕ 4 are fixed at ϕ 2 ≃ π/2 and ϕ 3 ≃ ϕ 4 ≃ 0. The experimental probabilities were calculated from the measured post-selected coincidence events with μ ≃ 367 as shown in Fig. 4a, b.In addition, the expected limit was obtained by calculating Eq. ( 1) using the sixteen probabilities with experimentally obtained non-ideal visibilities, and the SQL and the HS are also drawn by calculating 1=ð ffiffiffiffiffiffi ffi μN p Þ and 1=ð ffiffiffi μ p NÞ, respectively, with μ ≃ 367 in Fig. 4c.See Methods for error analysis on our experimental results.Note that the error bars on the standard deviation in Fig. 4c lie below the HS.It is mainly from the mismatch between experimental results and theoretically calculated probability functions in Figs.4a, b, and we attribute this to fluctuation on phases due to a few km long optical fibers (~3 km) and the fact that the limited number of samples are used for estimation (μ ≃ 367) 22 .Our experimental results clearly show that our strategy can achieve sensitivity beyond the SQL, and this can be further improved up to the HS with better visibility.We can also see that the obtained standard deviations are close to the HS in Fig. 4c.
In our experiment, we used a post-selection technique and did not include experimental imperfections such as photonic losses and lack of high-efficiency photon number resolving detectors.However, verifying the proof-of-concept of quantum enhancement is not affected by the post-selection technique, which has been used in most experiments for quantum-enhanced parameter metrology 21,[25][26][27][28][29] .Moreover, it is possible to achieve sensitivity beyond the SQL by using state-of-theart high-efficiency photon number resolving detectors 22,30,31 .See Supplementary Note for an analysis that takes into account all losses in our experiment setup.
We now consider a generalization of our scheme with Eq. ( 3) to Nphoton and d unknown phases to estimate an average phase φ = P d j = 1 ϕ j =d as shown in Fig. 5.The extended probe state in Eq. ( 3) can be described differently as follows: For example, Φ N 1,2 E denotes the N-photon two-mode polarizationentangled states between node 1 and node 2, i.e., , meaning that N/2 photons are sent to each node, respectively 32,33 .Here, N is assumed to be even.Then one can calculate the Fisher information matrix for Ψ N d E as follows:

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undergoes phase shifts at each node with a combination of two QWPs, and a HWP.In each node, σ x measurement is performed using a set of a HWP with an optic axis angle of 22.5 ∘ and a PBS.Then, the two-photon coincidence counts are measured using superconducting nanowire single-photon detectors (SNSPDs).b A polarization is generated from a Sagnac-interferometer-based Bell source 26,27 .Then, Ψ 2 Then the sensitivity bound can be calculated from Eq. ( 7) as Δ 2 φ = 1=N 2 , which is the HS of N photons.We note that the HS of N photons can be described using the average number of photons per mode, denoted as n = N/d, i.e., Δ 2 φ = 1=N 2 = 1=ðn 2 × d 2 Þ.It is noteworthy that our proposed protocol implies the HS for both the average photon number, n, and the number of unknown phases, d, which is desirable for distributed quantum sensing 16,19 .Moreover, note that in our scenario, it does not require the photon number N to be equal to or larger than the number of unknown phases d.It means that one can estimate the average of d unknown phases with N photons even for N < d.See Supplementary Note for detailed calculation on the generalization of our scheme.

Discussion
In summary, we experimentally demonstrated the distributed quantum phase sensing among four nodes with 2.2 dB sensitivity enhancement over the SQL.In our work, we estimated an average of four unknown phases located in distant nodes, which are separated by 3 km away from the central node.We theoretically showed that in our proposed scheme, one can achieve the HS even when the number of photons is less than the number of unknown phases.Moreover, we proposed a generalization of our scheme for estimating spatially distributed d phases using N photons with sensitivity of the HS.In addition, by utilizing the multiple passes of the phase shifter, our proposed scheme can estimate a linear global function of the distributed phases with the HS 21,22,34 .We believe that our results can provide helpful guidelines for a practical demonstration of spatially distributed quantum sensing by increasing the number of nodes consisting of the distributed sensor network.In addition, by exploiting the state-of-the-art long-distance entanglement distribution technique through deployed optical fiber networks [35][36][37] , our distributed quantum sensing scheme can be demonstrated in deployed optical fiber networks.It is also intriguing to combine distributed quantum sensing with simultaneous estimation of multiple parameters at a single location [26][27][28][29][38][39][40] by considering a scenario in which one finds an optimal strategy for estimating spatially  distributed multiple parameters when each node has more than a single parameter to be estimated.

Theoretical calculation of quantum Fisher information matrix and Δ 2 φ
We theoretically analyzed the ideal sensitivity limit of our scheme by calculating the weak form of CRB.taking into account {P l }, one can obtain the sensitivity bound from the Fisher information matrix F, which can be calculated as follows: Here, we can see the non-zero off-diagonal terms between the two adjacent nodes.Then the uncertainty bound of φ can be calculated to be Δ 2 φ = 1=4 by using Eq. ( 1) with α = (1/4, 1/4, 1/4, 1/4).Note that the sensitivity bound of 1/4 is equal to the HS 1/N 2 when N=2, and here the SQL is defined as 1/N = 1/2.

Error analysis
The standard deviation of estimated phase φest was obtained from experimentally obtained {P l } with μ measurements s times using MLE.
In our experiments, about 36,700 coincidence counts are divided into 100 groups for each phase shift.The error bar of standard deviation is obtained from an approximation δðΔ φÞ = Δ φ= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðs À 1Þ p with s = 100 groups 21,22 .

Fig. 1 |
Fig. 1 | Scheme for estimating spatially distributed multiple phases.Multiple unknown phases ϕ = {ϕ 1 , ϕ 2 , …, ϕ d } are located in each node.The goal of distributed sensing is to estimate the linear combination of the multiple unknown phases, φ, with a quantum-enhanced sensitivity.

Fig. 2 | 4 E
Fig. 2 | Experimental scheme.a Experimental setup for demonstrating our distributed quantum sensing scenario.The entangled probe state Ψ 2 4 E at the central node is sent to each node via 3 km fiber spools.Then, Ψ 2 4

Fig. 3 |
Fig.3| Experimental results on interference fringes between the node 1 and the node 2. The distance between the two nodes is 6 km (fiber spool).a, b Two-photon coincidence counts are obtained when the phase encoding φ 2 ½0,2π.Each surface represents a theoretical model P + + 12 ,P + À 12 ,P À + 12 , and P ÀÀ 12 in Eq. (5), respectively, and dots correspond to our experimental data.Note that the error bars are smaller than the markers.See Supplementary Note for the experimental interference fringe results that are obtained by scanning all phases.

Fig. 4 |
Fig. 4 | Experimental results on the estimated phase φest and the standard deviation Δ φest .a, b Experimentally obtained interference fringes of sixteen detection probabilities versus φ.Error bars are smaller than the markers.c The purple solid line represents the corresponding weak form of CRB value.The purple shaded area corresponds to 95% confidence regions, which are obtained from the standard deviation of fitting parameters.Cyan and red dashed lines correspond to the SQL and the HS, respectively.The error bars represent the standard deviation with s = 100 groups.See Methods.Detailed information on φest and Δ φest is provided in Supplementary Note.

Fig. 5 |
Fig. 5 | Generalized distributed sensing scheme with the Ψ N d E states.Probe state Ψ N d E are directed to two nodes out of d nodes, which is the superposition of multiphoton two-mode entangled states.The goal is to estimate the linear combination of distributed d unknown phases φ.